Rational Curves and (0,2)-Deformations

TL;DR

This paper investigates the discrepancy in counting (0,2)-deformation moduli between orbifold conformal field theories and their resolutions, attributing differences to worldsheet instanton corrections from rational curves.

Contribution

It analyzes rational curves on orbifold resolutions to identify contributions to worldsheet instanton corrections, revealing that smooth rational curves do not account for all discrepancies.

Findings

Irreducible toric rational curves explain some of the discrepancy.

Not all instanton corrections are from smooth rational curves.

Additional corrections must come from other types of rational curves.

Abstract

We compare the count of (0,2)-deformation moduli fields for N=(2,2) conformal field theories on orbifolds and sigma-models on resolutions of the orbifold. The latter involves counting deformations of the tangent sheaf. We see there is generally a discrepancy which is expected to be explained by worldsheet instanton corrections coming from rational curves in the orbifold resolution. We analyze the rational curves on the resolution to determine such corrections and discover that irreducible toric rational curves account for some, but not all, of the discrepancy. In particular, this proves that there must be worldsheet instanton corrections beyond those from smooth isolated rational curves.